Higher-spin Realisations of the Bosonic String

TL;DR

This paper explores higher-spin realizations of the bosonic string through linearising and embedding it into various $W$-algebra-based string theories, extending previous work on $W_3$ to $W_{2,4}$ and $W_{2,6}$ models.

Contribution

It introduces new embeddings of the bosonic string into $W_{2,4}$ and $W_{2,6}$ string theories by linearising their $W$-algebras, including a novel linearisation of $W_{2,6}$ at a specific central charge.

Findings

Linearisation of $W_{2,4}$ is known and applicable for all central charges.

Bosonic string can be embedded in critical and non-critical $W_{2,4}$ strings.

Linearisation of $W_{2,6}$ is achieved only at central charge $c=390$, enabling its embedding.

Abstract

It has been shown that certain $W$ algebras can be linearised by the inclusion of a spin--1 current. This provides a way of obtaining new realisations of the $W$ algebras. Recently such new realisations of $W_3$ were used in order to embed the bosonic string in the critical and non-critical $W_3$ strings. In this paper, we consider similar embeddings in $W_{2,4}$ and $W_{2,6}$ strings. The linearisation of $W_{2,4}$ is already known, and can be achieved for all values of central charge. We use this to embed the bosonic string in critical and non-critical $W_{2,4}$ strings. We then derive the linearisation of $W_{2,6}$ using a spin--1 current, which turns out to be possible only at central charge $c=390$. We use this to embed the bosonic string in a non-critical $W_{2,6}$ string.